Problem 117 - Chemistry The Central Science

Question

Cytochrome, a complicated molecule that we will represent as \(\mathrm{CyFe}^{2+},\) reacts with the air we breathe to supply energy required to synthesize adenosine triphosphate (ATP). The body uses ATP as an energy source to drive other reactions. (Section 19.7) At pH 7.0 the following reduction potentials pertain to this oxidation of \(\mathrm{CyFe}^{2+}:\) $$ \begin{aligned} \mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q)+4 \mathrm{e}^{-} \longrightarrow & 2 \mathrm{H}_{2} \mathrm{O}(l) & & E_{\mathrm{red}}^{\circ}=+0.82 \mathrm{~V} \\\ \mathrm{CyFe}^{3+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{CyFe}^{2+}(a q) & & E_{\mathrm{red}}^{\mathrm{o}}=+0.22 \mathrm{~V} \end{aligned} $$ (a) What is \(\Delta G\) for the oxidation of \(\mathrm{CyFe}^{2+}\) by air? (b) If the synthesis of \(1.00 \mathrm{~mol}\) of ATP from adenosine diphosphate (ADP) requires a \(\Delta G\) of \(37.7 \mathrm{~kJ},\) how many moles of ATP are synthesized per mole of \(\mathrm{O}_{2}\) ?

Step-by-Step Solution

Answer

The oxidation of CyFe²⁺ by air has a ∆G of -230.36 kJ/mol. Approximately 6.11 moles of ATP are synthesized per mole of O₂ in this reaction.

Step 1: Half-cell reactions

We are given two half-cell reactions: 1. \(\mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q)+4 \mathrm{e}^{-} \longrightarrow 2 \mathrm{H}_{2}\mathrm{O}(l)\) with \(E_{\mathrm{red}}^{\circ}=+0.82 \mathrm{V}\) 2. \(\mathrm{CyFe}^{3+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{CyFe}^{2+}(a q)\) with \(E_{\mathrm{red}}^{\circ}=+0.22 \mathrm{V}\) The first reaction is a reduction reaction, and the second one is an oxidation reaction. **Step 2: Write the overall balanced redox reaction**

Step 2: Overall balanced redox reaction

Combine the half-cell reactions to obtain the overall balanced redox reaction: \(\mathrm{O}_{2}(g) + 4 \mathrm{H}^{+}(a q) + 4 \mathrm{CyFe}^{2+}(a q) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) + 4 \mathrm{CyFe}^{3+}(a q)\) There are four electrons exchanged in this reaction, and the stoichiometry between O₂ and CyFe²⁺ is one-to-four. **Step 3: Calculate E° of the reaction**

Step 3: Calculate E° of the reaction

The standard cell potential E° for this reaction can be computed using the Nernst equation: \(E°_\text{cell} = E°_\text{red} - E°_\text{ox}\) Substitute the given values: \(E°_\text{cell} = (0.82 \, \text{V}) - (0.22 \, \text{V}) = 0.60 \, \text{V}\) **Step 4: Calculate ∆G for the reaction**

Step 4: Calculate ∆G for the reaction

Use the Nernst equation to find ∆G: \(\Delta G = -nFE°_\text{cell}\) where - n = the number of electrons exchanged in the balanced equation, - F = Faraday's constant (96,485 C/mol), and - E° = standard cell potential. Substitute the values: \(\Delta G = - (4\,\text{mol})(96,485 \frac{\text{C}}{\text{mol}})(0.60\text{V}) \Longrightarrow \Delta G = -230,364\text{J}\) So, \(\Delta G = -230.36\text{kJ/mol}\) **Step 5: Calculate the number of moles of ATP synthesized per mole of O₂**

Step 5: Number of moles of ATP synthesized per mole of O₂

The ∆G for the synthesis of 1 mol of ATP is given as 37.7 kJ. To find the number of moles of ATP synthesized per mole of O₂, we'll divide the ∆G for the oxidation of CyFe²⁺ (Step 4) by the ∆G for the synthesis of 1 mol of ATP: \(\text{Moles of ATP synthesized per mole of O₂} = -\frac{\Delta G_\text{oxygen}}{\Delta G_\text{ATP}} \Longrightarrow -\frac{-230.36 \,\text{kJ/mol}}{37.7 \,\text{kJ/mol}} \) \(\text{Moles of ATP synthesized per mole of O₂} \approx 6.11\) So, approximately 6.11 moles of ATP are synthesized per mole of O₂ in this reaction.

Key Concepts

Cytochrome

Cytochrome plays a crucial role in the electron transport chain within our cells. This complex protein contains a heme group, which is essential for its function in redox reactions. In simple terms, redox reactions involve the transfer of electrons between molecules. Cytochromes act as electron carriers that facilitate this transfer.
These proteins are pivotal in cellular respiration, a process that extracts energy from nutrients we consume. Within mitochondria, cytochromes transfer electrons from donors like

• NADH
• FADH₂

Ultimately passing them to oxygen, which combines with protons to form water. This process is not only key to generating energy but also ensures the efficient use of oxygen.
Interestingly, during these electron transfers, a proton gradient is created across the mitochondrial membrane. This gradient is later used to produce ATP, the primary energy currency of cells. Hence, cytochromes are vital in both energy generation and maintaining cellular homeostasis.

Adenosine Triphosphate (ATP)

Adenosine Triphosphate, commonly known as ATP, is often referred to as the energy currency of the cell. This molecule stores and provides energy for various cellular functions.
The structure of ATP includes an adenosine molecule bonded to three phosphate groups. These phosphate bonds are highly energetic. When one of these bonds breaks, ATP is converted to Adenosine Diphosphate (ADP), releasing energy in the process.
Cells utilize this released energy for:

• Muscle contraction
• Nerve impulse propagation
• Protein synthesis

Additionally, in the context of the problem presented, ATP is synthesized from ADP through the energy supplied by redox reactions. The synthesis requires a specific amount of energy, exttt{37.7 kJ/mol}, making ATP production efficient yet regulated. In essence, ATP serves as the bridge between energy-yielding reactions and energy-consuming processes in cells.

Standard Cell Potential

Standard Cell Potential exttt{(E^{ ext{°}})} is a measure of the potential difference between two electrodes in a cell under standard conditions exttt{(1M concentration, 1 atm pressure, and 25°C temperature)}. It reveals the cell's ability to push electrons through a circuit.
This potential is determined by the half-cell reactions occurring in the electrochemical cell. Each half-cell has a specific standard reduction potential, denoted as exttt{E^{ ext{°}}_{ ext{red}}}. For example, in the exercise, the reduction potential of oxygen is higher exttt{(+0.82 V)} compared to cytochrome exttt{(+0.22 V)}. Having a positive cell potential indicates a spontaneous redox reaction.
To find the standard cell potential for the overall reaction, subtract the lower reduction potential exttt{(E^{ ext{°}}_{ ext{ox}})} from the higher one:\[E^{ ext{°}}_{ ext{cell}} = E^{ ext{°}}_{ ext{red}} - E^{ ext{°}}_{ ext{ox}}\]This calculation gives exttt{0.60 V for the reaction}, indicating the intrinsic ability of the system to generate electrical energy, which can be utilized in various bioenergetic processes.